Light-Front Quantization with Explicit Lorentz Symmetry for Yukawa Model

Jerzy A Przeszowski,Jan Żochowski

doi:10.1007/s00601-014-0926-z

Abstract

The Dirac method for constrained systems is incomplete for the light-front (LF) quantization of the Yukawa model in D = 1 + 1 dimensions. A novel quantization procedure is proposed, where one obtains the LF commutator and anti-commutators directly from the Heisenberg equations generated by P+, which is a kinematical operator. By adding the general assumptions on the quantum field theory, one evalutes 2-point Wightman functions for a free field case. The Lorentz symmetry is manifest at every step of this novel LF procedure. The Gaussian effective potential is defined with the point-splitting regularization with a space-like separation. The optimum values of the mass parameters are regularization independent.

Highlights

The standard LF quantization [1,2,3] is the canonical quantization procedure for constrained systems. As it is shown in “Appendix B”, the Dirac procedure [4,5,6] is incomplete at the LF hypersurface for fermion fields
We may omit the classical canonical structure and start at the quantum level with the Heisenberg equations with P+ operator. This will allow us to read out the commutation relations for these fields, which are canonical at the LF hypersurface
By implementing some general properties of the quantum field theory we may evaluate the Wightman functions for the canonical fields and for those fields, which are not canonical. We illustrate this procedure by considering the Yukawa model in D = 1 + 1 dimensions, where the Lorentz invariant Lagrangian density is m2 φ2 2

Summary

Novel LF Quantization

The standard LF quantization [1,2,3] is the canonical quantization procedure for constrained systems. We may omit the classical canonical structure and start at the quantum level with the Heisenberg equations with P+ operator This will allow us to read out the commutation relations for these fields, which are canonical at the LF hypersurface. By implementing some general properties of the quantum field theory we may evaluate the Wightman functions for the canonical fields and for those fields, which are not canonical We illustrate this procedure by considering the Yukawa model in D = 1 + 1 dimensions, where the Lorentz invariant Lagrangian density is. From these relations we directly read out the non-vanishing LF canonical (anti)-commutators ψ+(x+, x−), ψ+† (x+, y−) In this model φ, ψ+, ψ+† are the LF canonical quantum fields.

Volterra Equations for Wightman Functions

Gaussian Effective Potential

Conclusions and Prospects